Calculus Examples

$\ln (x - 3) + \ln (2 x - 1) = \ln (3)$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln ((x - 3) (2 x - 1)) = \ln (3)$

Step 1.2

Expand $(x - 3) (2 x - 1)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$\ln (x (2 x - 1) - 3 (2 x - 1)) = \ln (3)$

Step 1.2.2

Apply the distributive property.

$\ln (x (2 x) + x \cdot - 1 - 3 (2 x - 1)) = \ln (3)$

Step 1.2.3

Apply the distributive property.

$\ln (x (2 x) + x \cdot - 1 - 3 (2 x) - 3 \cdot - 1) = \ln (3)$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$\ln (2 x \cdot x + x \cdot - 1 - 3 (2 x) - 3 \cdot - 1) = \ln (3)$

Step 1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.3.1.2.1

Move $x$ .

$\ln (2 (x \cdot x) + x \cdot - 1 - 3 (2 x) - 3 \cdot - 1) = \ln (3)$

Step 1.3.1.2.2

Multiply $x$ by $x$ .

$\ln (2 x^{2} + x \cdot - 1 - 3 (2 x) - 3 \cdot - 1) = \ln (3)$

Step 1.3.1.3

Move $- 1$ to the left of $x$ .

$\ln (2 x^{2} - 1 \cdot x - 3 (2 x) - 3 \cdot - 1) = \ln (3)$

Step 1.3.1.4

Rewrite $- 1 x$ as $- x$ .

$\ln (2 x^{2} - x - 3 (2 x) - 3 \cdot - 1) = \ln (3)$

Step 1.3.1.5

Multiply $2$ by $- 3$ .

$\ln (2 x^{2} - x - 6 x - 3 \cdot - 1) = \ln (3)$

Step 1.3.1.6

Multiply $- 3$ by $- 1$ .

$\ln (2 x^{2} - x - 6 x + 3) = \ln (3)$

Step 1.3.2

Subtract $6 x$ from $- x$ .

$\ln (2 x^{2} - 7 x + 3) = \ln (3)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$2 x^{2} - 7 x + 3 = 3$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Subtract $3$ from both sides of the equation.

$2 x^{2} - 7 x + 3 - 3 = 0$

Step 3.2

Combine the opposite terms in $2 x^{2} - 7 x + 3 - 3$ .

Tap for more steps...

Step 3.2.1

Subtract $3$ from $3$ .

$2 x^{2} - 7 x + 0 = 0$

Step 3.2.2

Add $2 x^{2} - 7 x$ and $0$ .

$2 x^{2} - 7 x = 0$

Step 3.3

Factor $x$ out of $2 x^{2} - 7 x$ .

Tap for more steps...

Step 3.3.1

Factor $x$ out of $2 x^{2}$ .

$x (2 x) - 7 x = 0$

Step 3.3.2

Factor $x$ out of $- 7 x$ .

$x (2 x) + x \cdot - 7 = 0$

Step 3.3.3

Factor $x$ out of $x (2 x) + x \cdot - 7$ .

$x (2 x - 7) = 0$

Step 3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$2 x - 7 = 0$

Step 3.5

Set $x$ equal to $0$ .

$x = 0$

Step 3.6

Set $2 x - 7$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.6.1

Set $2 x - 7$ equal to $0$ .

$2 x - 7 = 0$

Step 3.6.2

Solve $2 x - 7 = 0$ for $x$ .

Tap for more steps...

Step 3.6.2.1

Add $7$ to both sides of the equation.

$2 x = 7$

Step 3.6.2.2

Divide each term in $2 x = 7$ by $2$ and simplify.

Tap for more steps...

Step 3.6.2.2.1

Divide each term in $2 x = 7$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2}$

Step 3.6.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.6.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.6.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2}$

Step 3.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{2}$

Step 3.7

The final solution is all the values that make $x (2 x - 7) = 0$ true.

$x = 0, \frac{7}{2}$

Step 4

Exclude the solutions that do not make $\ln (x - 3) + \ln (2 x - 1) = \ln (3)$ true.

$x = \frac{7}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{2}$

Decimal Form:

$x = 3.5$

Mixed Number Form:

$x = 3 \frac{1}{2}$